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Distances in our world (classically at least) can be defined by considering our world to be a model for the Euclidean normed three dimensional space. But how can we set the notion of time on a firm footing? What are the first principles here?

Because almost everywhere in physics, we want to see how a system evolves with “time”. But when I think about it, I can’t find a clear understanding of what time is.

I’m not talking of “the arrow of time”, just time.

Before relativity also, physicists worked with time rates. But how can we be sure that a ticking device is measuring time linearly? As far as I can see, there must be a guaranteed (local, for relativity) existence at every place in the universe of some time keeping mechanism which is infinitely accurate and can be accessed by the observer to read off the time differences between events he observes.

What guarantees such existence?

Basically what I’m asking is against what the rate of time flow is measured. Or asking it is nonsense?

Is there some different way to firmly define this notion?

This is NOT the same question as the marked question, here.

Here, I ask how do we assure that a time keeping device ticks at “equal” intervals? How do we measure these “equal intervals” with? We surely can’t rely on our perception for it.

It is much like (to me) the problem of defining temperature in classical thermodynamics. We have to resort to statistical theory for a firm definition of it. We just can’t say that temperature is proportional to the height of mercury column in a thermometer. Because we’ll, it just isn’t...

Dale
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Atom
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  • https://arxiv.org/abs/0708.2743 https://arxiv.org/abs/0805.4452 – Count Iblis Aug 03 '19 at 19:01
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    Possible duplicate of What is time, does it flow, and if so what defines its direction? – John Rennie Aug 03 '19 at 19:03
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    The rate or time flow is measured against changes in the world around us that we can observe. The official SI standard is based on a caesium atom's ground-state electron transition frequency, because it is considered objective, universal and accessible (fairly "easy" to verify anywhere). But you could measure time according to a ticking pendulum, a revolution of the Sun on our sky or a decaying apple tree. Theoretically, if time suddenly slowed down to, say, half the current rate, you wouldn't be able to know, because everything you measure time with would likewise be slowed down similarly. – Steeven Aug 03 '19 at 19:10
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    @Steeven But how do we know that the Cs clock ticks at equal intervals? Or is it just that we define it to be at equal intervals? – Atom Aug 03 '19 at 19:21
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    @Atom That's the point. We don't. But we have to pick something as our reference to measure everything else against. If that reference itself is varying - if the Cs atom frequency suddenly changed on all Cs atoms on the planet - then we might notice because everything else, all clocks on earth, that have been set according to this reference still would agree with each other but suddenly not to the reference. But if other types of changes will or has happened, like if time changes universally, then we simply wouldn't be able to know because our reference would change along. – Steeven Aug 03 '19 at 19:30
  • @Steeven No, what I’m saying is that what if a Cs atom’s frequency is not perfectly (beyond our precision, of course) same as it was, say 10 years ago, as it is now. How can we be sure about this? By comparing it with other Cs atoms? – Atom Aug 03 '19 at 19:36
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    @Atom I'm not an expert, so anyone should correct me if I'm wrong, but ultimately, the idea that all Cs-133 atoms transition at the same rate at different points in space and time is an empirical matter. If it turns out that on some extreme level of precision the Cs atoms' frequencies are not the same, or if it turns out that time itself can't be parametrized by a single number, we will need need to devise a new theory to accommodate that. I think it comes down to the fact that all successful theories we know of assume time can have an ideal standardization... – Maximal Ideal Aug 03 '19 at 22:20
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    @Atom ...therefore, I think what people are trying to say is this: If all matter & energy follows such and such theory, then time can be standardized according to Cs-133 atoms. The problem, however, is that we can never know if all matter & energy follows our theories to all possible precision levels, so we are left to pursue empirical endeavors. I'm not sure if this helps, or if I'm totally off-base. – Maximal Ideal Aug 03 '19 at 22:24
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    FWIW, here's a couple of quotes from the first chapter of "Gravitation": (1) "Good clocks make spacetime trajectories of free particles look straight" and (2)"Time is defined to make motion look simple!" – Alfred Centauri Aug 03 '19 at 22:26
  • @Atom What I'm saying here is, empirical investigations (and the theories that ensue) show that there is a standard of time that linearly correlates with our neuronal activity and at the same time conveniently simplifies our equations of motion. (I'm ignoring GR here for simplicity, but a similar statement can hold there.) A deeper question of why this is the case (or whether this is even true at all levels of precision) would require a theory beyond what we currently understand. – Maximal Ideal Aug 04 '19 at 00:13
  • There is no proof at all that there exists time that is absolute, ie that the moment "now" or the moment "then" is definable and makes sense for two separate entities. Our, indeed that now and then are different. The only thing we know exists is relative time, how much time is between two events in a given reference frame. – Stian Aug 04 '19 at 06:16

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Although the answer given above is good, I want to add something for both the OP and future readers of this post.

In your question you state:

. . .But when I think about it, I can’t find a clear understanding of what time is.

Physics, as a mainstream science, doesn't seem to have much of an answer to questions about what time actually is. Physicists certainly wonder about the nature of time in their own spare time just as you seem to be wondering about it. But as far as normal experimental physics goes, being able to measure time with clocks is about as close to understanding time as we get. Scientists in general do not have much of any way to resolve issues about things above and beyond what they can measure.

the_photon
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    Note that the order of answers on stack exchange sites changes from time to time, so you should probably avoid phrases like "the answer given above." :) – D. Halsey Aug 03 '19 at 21:31
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Time (specifically proper time) is the thing measured by clocks. We can be sure that equal time intervals measured by clocks are equal because we have defined time to be the thing that clocks measure. If a clock measured twice as much of something and if the thing that clocks measure is defined to be time then what it measured was by definition twice as much time.

Now, there is no such thing as an ideal clock, but there are some clocks that agree with each other better than other clocks. These clocks that agree with each other well over a long time are used as the standard clocks, and groups of them (at least 3) are used together to detect when one starts to fail electronically or mechanically. Other clocks that are less stable are compared to the standard clocks and adjusted as needed. Clocks that don’t agree with the standard clocks are fixed or removed.

Dale
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  • But suppose the group of clocks we chose were such that they ticked in unison but the length between successive ticks increased quadratically with each number of tick. It would mean that the lifespan of humans is continuously increasing. Then what would we do? – Atom Aug 03 '19 at 20:13
  • Not only that, but it would also imply violation of time translation symmetry in many cases. Like I gave you the above example. – Atom Aug 03 '19 at 20:15
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    Such clocks would never be chosen as the reference standards because their disagreement rate would grow linearly in time and we already have clocks whose disagreement rate is constant in time. – Dale Aug 03 '19 at 20:44
  • Disagreement rate with what? I said that all the quadratic clocks ticked in unison always. – Atom Aug 03 '19 at 21:03
  • How could you conclude then which set of clocks, linear or quadratic, was correct? Cuz there’s no objective truth here. – Atom Aug 03 '19 at 21:05
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    @Atom If you chose to define time using the quadratic clocks, wouldn't Newton's laws of motion (along with electrodynamics and so on) have to be modified? Newton's first law says that objects (in an idealized scenario) that are not affected by forces trace out a straight-line. Based on observation (like with frictionless air-tracks), we see that there is a unique time standard that parametrizes all such straight-line motions linearly with respect to time. Secondly, your proposal would conflict with the Cesium-133 time standard (and the transition rates would be non-linear in your idea). – Maximal Ideal Aug 03 '19 at 22:04
  • “I said that all the quadratic clocks ticked in unison always”. Your hypothetical is getting progressively less believable. First, you have clocks that naturally violate the laws of physics by not being time translation invariant, second we are too stupid to notice that and pick them for our reference standard, and now these measuring devices (unlike any measurements of any kind ever) have no disagreements. At some point you need to bring your “what if” back to something relevant to the real world if you want to learn physics – Dale Aug 03 '19 at 23:33
  • @Atom I don't understand how your nonlinear time can be reconciled with the relativity of simultaneity. There is no universal "now", so it's not possible for clocks at different locations to maintain a synchronized nonlinear pattern for all observers. – PM 2Ring Aug 04 '19 at 01:57
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    @Atom Then the rate of human experience, aging, etc., would also increase quadratically and we wouldn't be able to observe the variation in ticks. If they didn't, we'd have picked that stuff as the clock in the first place. – chrylis -cautiouslyoptimistic- Aug 04 '19 at 03:36
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Much of the answer to your question is actually part of philosophy more than physics. "What is time?" is a famously difficult open ended question in philosophy. For example, philosophers will question whether time is discrete or continuous.

As for the practical scientist, what we observe is that the things we call clocks (i.e. hour glasses, watches, atomic clocks, etc.) behave in a way which is consistent with each other. If you look at the state of one clock, you can very reliably predict the state of another clock. And it seems the more "reliable" a clock is, the more predictable they are. Atomic clocks, for instance, are known to be remarkably consistent with respect to one another.

When we measure physical events, and reference this against the behaviors of these accurate clocks, we find that there is a very predictable pattern if we assume there is some variable, "time," which is ticked off linearly by these clocks. When we make this assumption, the laws of physics become "simple."

Thus we use it. Time can indeed be thought of as nothing more than a convenient analogy for physical behaviors. Or one may dabble into the philosophy to dig deeper.

Or, as an alternative, consider a reversed method of thinking. The thing scientists refer to as "time" is that which best fits the observations made about clocks. After all, science is an art of drawing conclusions about reality using data.

Cort Ammon
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  • Have you dived deeply into the philosophy behind this question? Do you think it is wrong if someone has not - hence why my answers are generally of unremarkable quality, because I'm doing wrong? – The_Sympathizer Aug 04 '19 at 03:03
  • @The_Sympathizer I have dived into that philosophy, and you will note that it did not make it into the answer other than to link the words of others. Time is a particularly tricky universal, which one reason its a question that remains open. As for how deep one may dive, I will readily admit that I have trouble appreciating the nuanced differences between McTaggart's A-series and B-series time, which suggests my own dive has been shallow by some standards. – Cort Ammon Aug 04 '19 at 19:59
  • Myself, i see more value in understanding that there is a point where these questions turn into philosophy, and you are welcome to cross that line or not. That helps you appreciate the assumptions which go into science, when to assume them, and when to question them. Dive when the water looks cool and refreshing; don't dive if its murky and stangant. That way, no matter what, it was a pleasant journey. – Cort Ammon Aug 04 '19 at 20:00
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In a comment you wrote:

But suppose that the group of clocks we chose were such that they ticked in unison but the length between successive ticks increased quadratically with each number of tick.

You seem to have a mental model of our universe as being some sort of simulation that’s running on a computer in some meta-universe, which has a notion of “meta-time” that you are thinking of as the “true” time. And yes, in this mental model it’s possible to imagine the operators of the simulation programming their computer to speed up or slow down the speed at which the simulation is running according to a quadratic function, or indeed any other nonlinear function. For example, they could decide that on meta-Mondays and meta-Wednesdays they will pause the simulation entirely between meta-noon and meta-1 pm to go eat meta-lunch. How would we humans know that this has happened? Well, we wouldn’t, because we are living inside the simulation, so while the simulation was paused, we would be paused too. When it resumes, we would resume too, and to us it would seem that the moment after meta-1 pm came immediately after the moment before meta-noon. From our point of view we would experience a sense of time flowing “linearly” and continuously, despite the discontinuity when you look at things in the meta-time parameter.

The key to resolving this difficulty, and to answering your “how can we be sure” question, is to recognize that these sorts of questions are meaningless within the framework of scientific theories and the scientific method. Specifically, your speculative theory about time flowing quadratically with respect to a more “true” time variable lacks the property of falsifiability, which is one of the basic requirements a scientific theory must satisfy:

I shall require that [the] logical form [of the theory] shall be such that it can be singled out, by means of empirical tests, in a negative sense: it must be possible for an empirical scientific system to be refuted by experience.

— Karl Popper, Popper 1959. p 19

The point is that science deals only with scientific questions, and what makes a question scientific is if it can be tested via experiment. Your question belongs to a class of “what if” questions that involve assumptions about how the universe “really” works, or what is happening “behind the curtains” or “under the hood” or “outside the simulation”, which invariably cannot be tested by any experiment, and therefore are not scientific questions. They may be debated by philosophers, or sometimes by physicists if they are in a certain mood, but they are not really mainstream physics questions, and physics has nothing useful to say about them.

(With that being said, such questions seem to be very common on this site, and are often interesting and entertaining to think about, so I don’t mean to suggest that there’s anything wrong with your asking about this.)

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